Let us first prove part (REF ).
Consider the category fibered in groupoids \(\widetilde{\mathcal {M}}_{G}\)  over the category of schemes, whose fiber over a scheme \(S\)  consists of families of quasi-stable curves \((\mathcal {C}\rightarrow S) \in \overline{\mathcal {M}}^{qs}_{g,n}(S)\)  together with an isomorphism \(\phi _{s}:G\xrightarrow{} G(\mathcal {C}_{s})\)  in \(\operatorname{\mathcal {QSG}}_{g,n}\)  for every geometric point \(s\rightarrow S\) , that are compatible with étale specializations in the following sense: for any étale specialization \(f\colon t\rightsquigarrow s\)  of geometric points, with induced morphism of graphs \(\pi _f:G(\mathcal {C}_{s})\rightarrow G(\mathcal {C}_t)\) , we require that \(\phi _t=\pi _f\circ \phi _s\) .
We claim that we have an isomorphism of categories fibered in groupoids
\(\widetilde{\mathcal {M}}_G\cong \prod _{v\in V(G)} \mathcal {M}_{h(v), \operatorname{val}(v)},\) 
which implies that \(\widetilde{\mathcal {M}}_G\)  is a smooth and irreducible algebraic stack of finite type over \(k\) .
Indeed, we have a functor \(\widetilde{\mathcal {M}}_G\rightarrow \prod _{v\in V(G)} \mathcal {M}_{h(v), \operatorname{val}(v)}\)  that sends \((\mathcal {C}\rightarrow S, \lbrace \phi _{s}\rbrace _s)\in \widetilde{\mathcal {M}}_{G}(S)\)  into the relative normalization \((\mathcal {C}^{\nu }\rightarrow S)\)  which is seen as an element of \(\prod _{v\in V(G)} \mathcal {M}_{h(v), \operatorname{val}(v)}(S)\)  via the compatible isomorphisms \(\lbrace \phi _s\rbrace _{s\in S}\) . Conversely, we have a functor \(\prod _{v\in V(G)} \mathcal {M}_{h(v), \operatorname{val}(v)}\rightarrow \widetilde{M}_G\)  that sends
\(\mathcal {D}\rightarrow S\in \prod _{v\in V(G)} \mathcal {M}_{h(v), \operatorname{val}(v)}(S)\)  into the family of quasi-stable curves \(\mathcal {C}\rightarrow S\)  of type \((g,n)\)  that is obtained from \(\mathcal {D}\rightarrow S\)  by gluing the pairs of marked sections of \(\mathcal {D}\rightarrow S\)  that correspond to the pairs of half-edges of \(G\)  that are exchanged by \(i_G\) , together with the family of compatible isomorphisms \(\lbrace \phi _s:G \xrightarrow{}G(\mathcal {C}_s)\rbrace _s\)  that is induced by the above gluing procedure. Clearly the two functors are inverse of each other and they define the isomorphism (REF ).

The algebraic stack \(\widetilde{\mathcal {M}}_G\)  comes with a universal family \(\pi _G:\widetilde{\mathcal {C}}_G\rightarrow \widetilde{\mathcal {M}}_G\)  of quasi-stable curves of type \((g,n)\)  together with compatible isomorphisms \(\lbrace \phi _s:G\xrightarrow{}G(\widetilde{\mathcal {C}}_{G,s})\rbrace _s\) , where \(s\)  runs over all geometric points of \(\widetilde{\mathcal {M}}_G\) , and \(\widetilde{\mathcal {C}}_{G,s}\)  is the geometric fiber of \(\pi _G\)  over the geometric point \(s\rightarrow \widetilde{\mathcal {M}}_G\) .
The category fibered in groupoids \(\widetilde{J}_{(G,D)}\) , endowed with its natural forgetful morphism \(\widetilde{\Psi }_{(G,D)}: \widetilde{J}_{(G,D)}\rightarrow \widetilde{\mathcal {M}}_G\) , can be naturally identified with the connected component of the Picard stack of the family \(\pi _G:\widetilde{\mathcal {C}}_G\rightarrow \widetilde{\mathcal {M}}_G\)  that parametrizes line bundles \(\mathcal {L}\)  on \(\widetilde{\mathcal {C}}_G\)  whose multidegree satisfies \(\underline{\deg }(\mathcal {L}_s)=\phi _s(D)\)  for any geometric point \(s\rightarrow \widetilde{\mathcal {M}}_G\) , where \(\mathcal {L}_s\)  is the restriction of \(\mathcal {L}\)  to the geometric fiber \(\widetilde{\mathcal {C}}_{G,s}\) . Hence, we deduce that \(\widetilde{J}_{(G,D)}\)  is a smooth and irreducible algebraic stack of finite type over \(k\) .
We now prove part (REF ). We begin by establishing several properties of the forgetful morphism \(\Phi _{(G,D)}:\widetilde{J}_{(G,D)}\rightarrow J_{(G,D)}\) .

 \(\Phi _{(G,D)}\)  is surjective and the fiber over a geometric point \(\operatorname{Spec}K \rightarrow J_{(G,D)}\)  is a torsor under the constant finite group scheme \(\operatorname{Aut}(G,D)_K\) .
Indeed, if \((C,L)\in J_{(G,D)}(K)\)  then the fiber of \(\Phi _{(G,D)}\)  over \((C,L)\)  is the set of all the isomorphisms \((G,D)\rightarrow (G(C),\underline{\deg }L)\) , which is non-empty because of our assumption that
\((C,L)\in J_{(G,D)}(K)\)  and it is clearly a torsor for \(\operatorname{Aut}(G,D)\) .

 \(\Phi _{(G,D)}\)  is representable, finite and unramified.
In order to prove this, consider the commutative diagram
\({\widetilde{J}_{(G,D)}@/^/[rrrd]^{\widetilde{\Psi }_{(G,D)}}@/_/[ddr]_{\Phi _{(G,D)}} [dr]^{F_{(G,D)}}& & & \\& J_{(G,D)}\times _{\mathcal {M}_G}\widetilde{\mathcal {M}}_G[rr]^{\overline{\Psi }_{(G,D)}}[d]^{\overline{\Phi }_G}@{}[rrd]|\square & & \widetilde{\mathcal {M}}_G [d]^{\Phi _G}\\& J_{(G,D)} [rr]_{\Psi _{(G,D)}}& & \mathcal {M}_G}\) 
where \(\mathcal {M}_G\)  is the locally closed (reduced) substack of \(\mathcal {M}_{g.n}^{qs}\)  whose \(k\) -points are \(C\in \mathcal {M}_{g,n}^{qs}(k)\)  such that \(G(C)\cong G\) , the morphism \(\Phi _G\)  forgets the isomorphisms \(\lbrace \phi _s\rbrace \) , the morphism \(\Psi _{(G,D)}\)  forgets the line bundle, the square is cartesian and the morphism \(F_{(G,D)}\)  is induced by the universal property of the fibered product.
Now, property (REF ) will follow from the following two properties:

\(\overline{\Phi }_{G}\)  is representable, finite and unramified.
Indeed, it is enough to show that \(\Phi _G\)  is representable, finite and unramified. It follows from (REF ) (and its proof) that \(\Phi _G\)  is a composition of clutching morphisms which are known to be representable, finite and unramified by [1]}.

\(F_{(G,D)}\)  is a representable closed embedding.
Indeed, the fibered product \(J_{(G,D)}\times _{\mathcal {M}_G}\widetilde{\mathcal {M}}_G\)  is the algebraic stack whose fiber over a scheme \(S\)  consists of \((\mathcal {C}\rightarrow S, \mathcal {L}, \lbrace \phi _{s}:G\xrightarrow{} G(\mathcal {C}_{s})\rbrace _s)\) , where
\((\mathcal {C}\rightarrow S, \lbrace \phi _{s}:G\xrightarrow{} G(\mathcal {C}_{s})\rbrace _s)\in \widetilde{\mathcal {M}}_G\)  and \(\mathcal {L}\)  is an admissible line bundle on \(\mathcal {C}\rightarrow S\)  such that \((G,D)\cong (G(\mathcal {C}_s),\underline{\deg }\mathcal {L}_s)\) .
This implies that \(\overline{\Psi }_{(G,D)}:J_{(G,D)}\times _{\mathcal {M}_G}\widetilde{\mathcal {M}}_G\rightarrow \widetilde{\mathcal {M}}_G\)  is the union of the connected components of the Picard stack of the family \(\pi _G:\widetilde{\mathcal {C}}_G\rightarrow \widetilde{\mathcal {M}}_G\)  that parametrize line bundles \(\mathcal {L}\)  on \(\widetilde{\mathcal {C}}_G\)  whose multidegree is such that there exists an automorphism \(\phi \)  of \(G\)  such that \((\phi _s)^{-1}(\underline{\deg }(\mathcal {L}_s))=\phi _*(D)\) 
for every geometric point \(s\rightarrow \widetilde{\mathcal {M}}_G\) . By what is proved in part (REF ), it follows that \(F_{(G,D)}\)  identifies \(\widetilde{J}_{(G,D)}\)  as one of the connected components of \(J_{(G,D)}\times _{\mathcal {M}_G}\widetilde{\mathcal {M}}_G\rightarrow \widetilde{\mathcal {M}}_G\) , namely the one corresponding to \(\phi =\operatorname{id}\) , and this implies that \(F_{(G,D)}\)  is a representable closed (and also open) embedding.


 \(\Phi _{(G,D)}\)  is flat.
This follows by applying the flatness criterion [2]} to the representable morphism \(\Phi _{(G,D)}\)  and using that \(\Phi _{(G,D)}\)  is surjective and finite (by (REF ) and (REF )) from the smooth and irreducible (by part REF ) algebraic stack \(\widetilde{J}_{(G,D)}\)  to the smooth and irreducible (by Proposition (REF ) (REF )) algebraic stack \(J_{(G,D)}\) .

By using the above properties of the morphism \(\Phi _{(G,D)}\) , we can now conclude the proof of part (REF ). Properties (REF ) and (REF ) and the first part of property (REF ) imply that \(\Phi _{(G,D)}\)  is a representable, surjective, finite and étale morphism. The second part of property (REF ) implies that \(\Phi _{(G,D)}\)  is a torsor under the constant finite group scheme \(\operatorname{Aut}(G,D)_k\) , which implies that \(\Phi _{(G,D)}\)  is a \(\operatorname{Aut}(G,D)\) -Galois morphism.
